Optimal. Leaf size=22 \[ \frac {2 (d \tan (a+b x))^{5/2}}{5 b d} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2607, 32} \[ \frac {2 (d \tan (a+b x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (d x)^{3/2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {2 (d \tan (a+b x))^{5/2}}{5 b d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 22, normalized size = 1.00 \[ \frac {2 (d \tan (a+b x))^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 45, normalized size = 2.05 \[ -\frac {2 \, {\left (d \cos \left (b x + a\right )^{2} - d\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 24, normalized size = 1.09 \[ \frac {2 \, \sqrt {d \tan \left (b x + a\right )} d \tan \left (b x + a\right )^{2}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 19, normalized size = 0.86 \[ \frac {2 \left (d \tan \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 18, normalized size = 0.82 \[ \frac {2 \, \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 100, normalized size = 4.55 \[ \frac {2\,d\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}\,\left (\cos \left (2\,a+2\,b\,x\right )-2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )+2\right )}{5\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )+6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )+10\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \sec ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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